Liberal Republican Senator Susan Collins of ME, who had previously expressed support for reopening the government without funding the wall, said she hoped that Mr. Trump's offer would "lead to constructive debate that will end this impasse".
The government shutdown is now the longest in USA history, going over the 30-day mark and threatening critical programs throughout the country.
Trump also said he was concerned about a new wave of immigrants moving north through Mexico toward the U.S. border.
"I've been talking about barriers for a long time", Clyburn said.
His plan included $800 million in humanitarian assistance; $805 million for security technology at Ports of Entry; 2,750 new border agents and law enforcement officials; 75 new immigration judge teams to address immigration cases. Democrats have rejected the proposal, and it appears unlikely to garner the 60 votes necessary to advance.
The proposal also did not strike a chord with prominent conservative commentator Ann Coulter, a fervent border wall advocate, one of those left-wing media is blaming for taunting Trump into a government shutdown in the first place.
Republican Senate Majority Leader Mitch McConnell welcomed the plan as a "bold solution", while a spokesman said McConnell would seek Senate passage of the proposal this week. "So if we grant citizenship to a BILLION foreigners, maybe we can finally get a full border wall". "This is not an amnesty bill", the vice president said. He has also offered a similar protection for those fleeing disaster zones, all in exchange for border wall money.
Trump had billed the plan as a compromise, but Democrats said it didn't go almost far enough, prompting a flurry of Sunday morning tweets.
In a White House address Saturday, Trump had proposed a reprieve on his attempts to end the DACA program and temporary protected status (TPS) for immigrants from some Latin American and African nations, in exchange for building hundreds of miles of barriers on the southern USA border and hiring thousands of new law enforcement agents to be deployed there.
The Trump administration had moved to end an Obama-era initiative known as Deferred Action for Childhood Arrivals, or DACA, which protects some immigrants brought to the USA illegally as children from being deported. Trump's proposal also includes a three-year extension of the Temporary Protected Status (TPS) that now lets people from certain countries stay in the United States if they left their homes to escape wars, disasters, or "other extraordinary and temporary conditions". He has refused to sign any spending bill without the funding. DACA and TPS were programs previously targeted by the Trump administration, despite bipartisan criticism.
The House Majority Whip responded to Trump's address, in a statement that said "until the President releases his hostages - federal workers and the American people - there will be no negotiation".
As the longest partial government shutdown in history drags on, the Senate will move on a measure this week the White House hopes will bring all sides to the table. Sunday's editorial explains why young people who came to the USA through the Obama-era program are in peril of being deported.
Democrats have said they don't want to reward Trump for what they say is taking 800,000 federal workers hostage to get his border wall, lest they give him an incentive to use the shutdown tactic again. The Democrats want to bring amendments or recommendations forward. He also said he was optimistic the plan would begin to gain traction after McConnell brings it to the Senate floor on Tuesday - though it's unclear it can pass.